Relating the modular Hamiltonian to two-point functions

TL;DR

This paper explores the relationship between the modular Hamiltonian and two-point functions in free scalar fields, providing new derivations and generalizations within quantum field theory.

Contribution

It establishes a direct connection between the modular Hamiltonian and two-point functions, extending previous formulas and applying to general CCR algebras.

Findings

Derived formulas relating modular Hamiltonian to two-point functions

Reproduced previous results using KMS condition

Generalized results to CCR algebras

Abstract

We consider the modular Hamiltonian associated to standard subspaces for a free scalar field in a globally hyperbolic spacetime in an arbitrary Gaussian state. We show how the modular Hamiltonian is related to the two-point function of the theory. For the restriction of the modular Hamiltonian to the subspace, we recover formulas that were obtained previously by Peschel, Casini and Huerta. We also show how the same results can be obtained more directly from the KMS condition, and generalize our results to general CCR algebras.